Average Error: 0.0 → 0.0
Time: 2.1s
Precision: 64
\[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.9189385332046730026078762421093415468931\]
↓
\[\mathsf{fma}\left(y - 1, x, \mathsf{fma}\left(0.5, -y, 0.9189385332046730026078762421093415468931\right)\right)\]
\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.9189385332046730026078762421093415468931
↓
\mathsf{fma}\left(y - 1, x, \mathsf{fma}\left(0.5, -y, 0.9189385332046730026078762421093415468931\right)\right)
double f(double x, double y) {
        double r31390 = x;
        double r31391 = y;
        double r31392 = 1.0;
        double r31393 = r31391 - r31392;
        double r31394 = r31390 * r31393;
        double r31395 = 0.5;
        double r31396 = r31391 * r31395;
        double r31397 = r31394 - r31396;
        double r31398 = 0.918938533204673;
        double r31399 = r31397 + r31398;
        return r31399;
}
↓
double f(double x, double y) {
        double r31400 = y;
        double r31401 = 1.0;
        double r31402 = r31400 - r31401;
        double r31403 = x;
        double r31404 = 0.5;
        double r31405 = -r31400;
        double r31406 = 0.918938533204673;
        double r31407 = fma(r31404, r31405, r31406);
        double r31408 = fma(r31402, r31403, r31407);
        return r31408;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.9189385332046730026078762421093415468931\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y - 1, x, \mathsf{fma}\left(0.5, -y, 0.9189385332046730026078762421093415468931\right)\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(y - 1, x, \mathsf{fma}\left(0.5, -y, 0.9189385332046730026078762421093415468931\right)\right)\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (* x (- y 1)) (* y 0.5)) 0.918938533204673))